局部同胚的动力学比较
Dynamical comparison for local homeomorphisms
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中文总结 AI 辅助
本文证明了紧可度量化空间上极小满射非单射局部同胚对应的Deaconu–Renault群胚的动力学比较性质，由此得到相关C*-代数的性质、验证了Matui AH猜想，还证明了极小群作用的经典小边界性质。
中文摘要 AI 辅助
我们证明了紧可度量化空间上具有有限勒贝格覆盖维数的极小满射非单射局部同胚对应的Deaconu–Renault群胚的动力学比较性质。作为推论，对应的C*-代数是满足UCT的Kirchberg代数，通过动力学方法重新得到了Carlsen–Thomsen的结果。在零维情形，我们利用Xin Li的最新突破，验证了这类群胚的Matui AH猜想。我们的证明结合了纯无限和稳定有限两种情形的技术：我们构造非阿贝尔自由群的部分作用作为合适的“大子群胚”，并使用Gardella–Geffen–Kranz–Naryshkin开发的悖论塔技术建立这些作用的比较性质。子群胚的边界由群胚版本的拓扑小边界性质控制，我们从单位空间的有限覆盖维数推导出该性质。作为副产品，我们证明了可数离散群在有限维紧可度量化空间上的极小作用的经典小边界性质，且无需任何自由性假设。
英文摘要
We prove dynamical comparison for Deaconu--Renault groupoids associated to minimal surjective non-injective local homeomorphisms of compact metrizable spaces with finite Lebesgue covering dimension. As a corollary, the associated $C^*$-algebras are UCT Kirchberg algebras, recovering results by Carlsen--Thomsen via dynamical methods. In the zero-dimensional case, our result also verifies Matui's AH-conjecture for these groupoids using a recent breakthrough of Xin Li. Our proof combines techniques from both the purely infinite and stably finite regimes: We construct partial actions of non-abelian free groups as suitable ``large subgroupoids'' and establish comparison properties for these using the paradoxical towers technique developed by Gardella--Geffen--Kranz--Naryshkin. The boundary of the subgroupoid is controlled by a groupoid version of the topological small boundary property which we deduce from finite covering dimension of the unit space. As a byproduct, we prove the classical small boundary property for minimal actions of countable discrete groups on finite-dimensional compact metrizable spaces without any freeness assumption.
